What is linear approximation?
Answer: f(x) ≈ f(a) + f'(a)(x − a) near x = a.
Explanation
The tangent line as an estimate; differentials dy = f'(x)dx.
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Calculus: Derivatives and Integrals (University)
ApplicationsSeriesIntegralsLimitsDerivative rules
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Applications
What is a related rates problem's general approach?
Answer: Write an equation relating the quantities, differentiate with respect to time using the chain rule, substitute known values, solve for the unknown rate.
Explanation
Draw a picture and identify what is constant.
State the Mean Value Theorem.
Answer: If f is continuous on [a,b] and differentiable on (a,b), there is a c in (a,b) with f'(c) = (f(b) − f(a)) / (b − a).
Explanation
Some tangent is parallel to the secant line.
What does the second derivative tell you about a graph?
Answer: Concavity: f'' > 0 concave up, f'' < 0 concave down; inflection points are where concavity changes.
Explanation
f'' = 0 alone does not guarantee an inflection point.
How do you find critical points and classify them?
Answer: Set f'(x) = 0 or undefined; use the first derivative test (sign change) or second derivative test (f'' > 0 min, f'' < 0 max).
Explanation
Endpoints must be checked for absolute extrema on a closed interval.
Series
Geometric series sum
Answer: ∑ arⁿ (n from 0) = a / (1 − r) when |r| < 1
Explanation
Diverges when |r| ≥ 1.
Taylor series of eˣ, sin x, cos x at 0
Answer: eˣ = ∑ xⁿ/n!; sin x = ∑ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!; cos x = ∑ (−1)ⁿ x²ⁿ/(2n)!
Explanation
All converge for every real x.
Ratio test
Answer: Let L = lim |aₙ₊₁ / aₙ|: converges absolutely if L < 1, diverges if L > 1, inconclusive if L = 1.
Explanation
Best for factorials and exponentials.
What is the nth-term test for divergence?
Answer: If lim aₙ ≠ 0, the series ∑aₙ diverges.
Explanation
Does not prove convergence when the limit is 0 (harmonic series).
Integrals
What is an improper integral and when does it converge?
Answer: An integral with an infinite limit or an unbounded integrand; it converges if the limit of the proper integrals exists and is finite.
Explanation
∫₁^∞ 1/xᵖ dx converges when p > 1.
Average value of a function on [a,b]
Answer: (1/(b−a)) ∫ₐᵇ f(x) dx
Explanation
The Mean Value Theorem for integrals says f takes this value somewhere on the interval.
Volume by cylindrical shells
Answer: V = 2π ∫ₐᵇ (radius)(height) dx, for rotation about a vertical axis.
Explanation
Useful when the washer method requires solving for x.
Volume by the disk/washer method
Answer: V = π ∫ₐᵇ [R(x)² − r(x)²] dx for rotation about the x-axis.
Explanation
Disk method when there is no inner radius.
How do you find the area between two curves?
Answer: ∫ₐᵇ [top(x) − bottom(x)] dx, splitting the interval where the curves cross.
Explanation
Or integrate with respect to y using right minus left.
Integration by parts formula
Answer: ∫ u dv = uv − ∫ v du
Explanation
Choose u by LIATE: logs, inverse trig, algebraic, trig, exponential.
How does u-substitution work?
Answer: Choose u = g(x) so that du = g'(x)dx appears in the integrand, rewrite in terms of u, integrate, substitute back.
Explanation
Change the limits of integration when doing a definite integral in terms of u.
Basic antiderivatives of eˣ, sin x, cos x, sec²x
Answer: ∫eˣ dx = eˣ + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C
Explanation
Always add the constant for indefinite integrals.
Power rule for integrals
Answer: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1
Explanation
∫ 1/x dx = ln|x| + C.
State the Fundamental Theorem of Calculus (both parts).
Answer: Part 1: d/dx ∫ₐˣ f(t)dt = f(x). Part 2: ∫ₐᵇ f(x)dx = F(b) − F(a) where F' = f.
Explanation
Links derivatives and integrals.
Limits
What is the limit definition of the derivative?
Answer: f'(x) = lim(h→0) [f(x+h) − f(x)] / h
Explanation
The slope of the tangent line at x.
What is L'Hôpital's rule?
Answer: For limits of the form 0/0 or ∞/∞, lim f/g = lim f'/g' if the latter exists.
Explanation
Only apply to indeterminate forms.
When is a function continuous at x = a?
Answer: f(a) is defined, lim(x→a) f(x) exists, and the limit equals f(a).
Explanation
Differentiable implies continuous, not the reverse.
Derivative rules
How does implicit differentiation work?
Answer: Differentiate both sides with respect to x, applying the chain rule to y terms (multiply by dy/dx), then solve for dy/dx.
Explanation
Used for curves like x² + y² = 25.
Derivatives of inverse trig functions
Answer: d/dx arcsin x = 1/√(1−x²); d/dx arctan x = 1/(1+x²)
Explanation
d/dx arccos x = −1/√(1−x²).
Derivatives of eˣ and ln x
Answer: d/dx eˣ = eˣ; d/dx ln x = 1/x
Explanation
General: d/dx aˣ = aˣ ln a; d/dx log_a x = 1/(x ln a).
Derivatives of sin, cos, tan
Answer: d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec²x
Explanation
Also: d/dx sec x = sec x tan x; d/dx cot x = −csc²x.
Chain rule
Answer: d/dx f(g(x)) = f'(g(x)) · g'(x)
Explanation
Differentiate the outside, keep the inside, multiply by the derivative of the inside.
Quotient rule
Answer: (f/g)' = (f'g − fg') / g²
Explanation
'Low d-high minus high d-low over low squared.'
Product rule
Answer: (fg)' = f'g + fg'
Explanation
'First times derivative of second plus second times derivative of first.'
Power rule
Answer: d/dx [xⁿ] = n·xⁿ⁻¹
Explanation
Works for any real n, including negatives and fractions.
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